We calculate the evolution of zero-metallicity Population III (Pop III) stars whose mass grows from the initial mass of ~1 M ☉ by accreting the surrounding gases. Our calculations cover whole evolutionary stages from the pre-main sequence, via various nuclear burning stages, through the final core-collapse or pair-creation instability phases. We adopt two different sets of stellar mass accretion rates as our fiducial models. One is derived from a cosmological simulation of the first generation (PopIII.1) stars, and the other is derived from a simulation of the second generation stars that are affected by radiation from PopIII.1 stars. The latter represents one case of PopIII.2 stars. We also adopt additional models that include radiative feedback effects. We show that the final mass of Pop III.1 stars can be as large as ~1000 M ☉, beyond the mass range (140-300 M ☉) for the pair-instability supernovae. Such massive stars undergo core-collapse to form intermediate-mass black holes, which may be the seeds for merger trees to supermassive black holes. On the other hand, Pop III.2 stars become less massive (40-60 M ☉), being in the mass range of ordinary iron core-collapse stars. Such stars explode and eject heavy elements to contribute to chemical enrichment of the early universe as observed in the abundance patterns of extremely metal-poor stars in the Galactic halo. In view of the large range of possible accretion rates, further studies are important to see if these fiducial models are actually the cases.